Symmetry and the Local Field Response in Photonic Crystals

Singular states of photonic crystal local optical field are described based on group theoretic treatments of the fundamental system symmetries expressed through local representations of the singular character of the Bloch modes. The local field response is illustrated, in a purely two-dimensional system for clarity, by an analysis of the local polarization state. The fundamentals of group theory are introduced through the group representation, whose partner functions may be used to express the photonic crystal eigenmodes. We have extended the eigenmode k-group treatment to the derivation of the irreducible matrix representations of the transformation operators of the eigenmode space k-group, which may be applied with respect to any symmetry axis. This permits a comprehensive discussion of the local polarization state, and its symmetry transformation properties, through the set of crystallographic orbits. Transformation relations permit the nature and location of polarization singularities to be identified by the sites' Wyckoff positions. Confirmation with analytic mode determination is presented.